Assessment of Voice Coil Peak Displacement XMAX 02

8/9/2019 Assessment of Voice Coil Peak Displacement XMAX 02

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Assessment of Voice Coil Peak Displacement X max

Wolfgang Klippel

Klippel GmbHDresden, 01277, Germany

www.klippel.de

ABSTRACTThe voice coil peak displacement X max is an important driver parameter for assessing the maximal acoustic output at lowfrequencies. The method defined in standard AES 2-1984 is based on a harmonic distortion measurement, which does notgive a definite and meaningful value of X max. After a critical review of this performance-based technique, an amendment of this method is suggested by measuring both harmonic and modulation distortion in the near field sound pressure using a twotone excitation signal. Alternatively, a parameter-based method is developed giving more detailed information about thecause of the distortion, limiting and defects. The relationship between performance-based and parameter-based methods isdiscussed, and both techniques are tested with real drivers.


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KLIPPEL Assessment of Voice Coil Peak Displacement X max

3

K L I P P E L

0 ,0

0 ,5

1 ,0

1 ,5

2 ,0

2 ,5

3 ,0

3 ,5

4 ,5

5 ,0

5 ,5

-1 5 -1 0 -5 0 5 1 0 1 5

B l

[N/A]

D i s p l a c e m e n t X [ m m ]

Fig. 2: Bl(x)-product versus voice coil displacement of thefictitious driver used in the simulation.

The force factor Bl(x) is not a constant parameter but a functionof voice coil displacement. The Bl(x) has a symmetrical bell-shaped form approaching zero for high displacement. A voice coilwith a height of 5 mm and a stray field outside the gap may causesuch a characteristic. The early strong decay of the Bl -curve istypical for a short voice coil overhang. Since the Bl(x)- curve is

perfectly symmetrical to x=0 the driver produces only third andother odd-order distortion.All other nonlinearities inherent in real drivers are neglected in thefictitious driver to keep our test case as simple as possible. Thuswe assume a linear suspension having a constant complianceCms(x)=const. and a voice coil inductance L e(x) that is independenton displacement. For the validity check those simplifications areadmissible because our goal is to find at least one case where theassumptions of the current definition fails.According to the measurement conditions defined in AES2-1984,we operate the driver in free air.

2.5 Simulation of Large Signal Behavior The fictitious driver can be precisely modeled by the nonlinear differential equation. Applying numerical integration [4] all thestate variables (current, displacement, etc.) and the acousticaloutput signal may be predicted for any input signal and may besubjected to a FFT analysis. Exciting the driver with a single toneat f 1=f s with varied terminal voltage U 1 Fig. 3 shows the totalharmonic distortion

%100*)(...)3()2()(

)(...)3()2(2

12

12

12

1

212121

Kf P f P f P f P Kf P f P f P d t ++++

+++=

of the current, sound pressure and displacement versus peak displacement X rms(f 1 ).

0 2 4 6 8 10 12 14 16 18 20010

20

30

40

50

60

70

80

voice coil peak displacement [ mm]

d t %

Fig. 3: Total harmonic distortion in current (dotted line), sound pressure (dashed line) and displacement (solid line) for a singleexcitation tone at f s versus voice coil peak displacement X

2.6 Applying X max definitionTo find the peak displacement X max according to the AES2-1984we have to search for peak displacement giving d t = 10 %. Usingthe total distortion in the input current we get a value X max about0.6 mm. That is a very small value compared to the voice coilheight of about 5 mm. No manufacturer would agree to specify theworking range of his loudspeaker to such a small signal domain. Inthis range of -0.6 mm < x < 0.6 mm the Bl(x) varies only by 5 %and the distortion in the radiated sound pressure is merely 2 %.Most likely the manufacturer would consider the alternativemethod. The total harmonic distortion in displacement remainsvery small and does not reach 10 % even if the coil is entirelyoutside the gap. Only common sense (but not the currentdefinition) prevents a manufacturer from setting the peak displacement X max to 19 mm and more.

2.7 X max from sound pressure distortionSome users modified the current X max definition and applied thethreshold of 10 % to the total harmonic distortion in the radiatedsound pressure. Usually this provides more reasonable estimates of

X max. However, we might get multiple values of X giving the samevalue of distortion. For example the fictitious driver provides threedifferent values 1.5 mm, 8 mm and 13.5 mm as candidates of X max.What is the right value to state?

2.8 What is wrong with the definition?Apparently the assumptions made in the current X max-definition are

not valid.First, the harmonic distortion in current and displacement at f s arenot in the same order of magnitude. The reason is quite simple.The amplitude of the fundamental component of the voice coilcurrent is minimal at the resonance frequency f S where theelectrical impedance is maximal.However, the generated harmonic components see much lower impedance at higher frequencies and therefore get boosted tohigher amplitude value. This effect of fundamentals suppressionis typical only for the voice coil current and is inherent neither todisplacement nor to sound pressure.There is also no simple relationship between harmonic distortionand peak displacement. Instead of a monotonically increase weobserve that the distortion stagnates at a relatively small valuegiving multiple values for X max. Fig. 4 shows the fundamental of the displacement versus terminal input voltage.


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4

KLIPPEL

0,0

2,5

5,0

7,5

10,0

12,5

0,0 2,5 5,0 7,5 10,0 12,5 15,0 17,5 20,0

Fundamental component| X ( f1, U1 ) |

X [ m m

] ( r m s

)

Voltage U1 [V]

46.9 Hz

Fig. 4: Amplitude of the displacement for a tone at f s versus inputvoltage U 1

For a 5-mm peak displacement, the instantaneous level of forcefactor Bl(x) reduces to 20 % and at 12 mm the force factor almostvanishes. Surprisingly, the motor works properly and we still getalmost a linear relationship between displacement and voltage.There are two reasons for that:

The electrical damping caused by Bl(x) 2 /Re decreasesand the mechanical Qms dominates the total dampingQ ts. The rising value of Q ts compensates for the reducedexcitation force F = Bl(x)i .

The 90-degree phase shift between the current i anddisplacement x at the resonance frequency still providesgood excitation conditions as shown in Fig. 5. Whenthe current is maximal the coil still remains in the gapand the instantaneous value of Bl(x) is also maximalthus producing a high driving force F = Bl(x)i and agood excitation of the system.

When the coil is outside the gap, we have a relatively high currentdetermined by the DC resistance Re, but when the coil movesthrough the gap some back EMF is generated, which produces thesmall dip in the current waveform.

-20

-15

-10

-5

0

5

10

15

20

0 ,0 0 0 0 ,0 05 0 ,0 1 0 0 ,0 15 0 ,0 20 0 ,0 25 0 ,0 30 0 ,0 3 5 0 ,0 40 0 ,0 45 0 ,0 50 0 ,0 5 5 0 ,0 60

Voice coil displacement X(t) vs time

X [ m m

]

Time [s]

X(t) I(t)

Fig. 5: Voice coil current i and displacement x versus time for anexcitation tone at resonance frequency f s.

Above and below the resonance frequency, variation of Bl(x) havea significant effect on the output. Fig. 6 shows the amplitude of voice coil displacement X rms as a function of frequency at variedinput voltage U 1.

KLIPPEL

0,0

2,5

5,0

7,5

10,0

12,5

Xrms

mm]

Frequency f1 [Hz]

100 1000

Voltage U 1

20

Fig. 6: Amplitude of Voice Coil Displacement of the fictitiousdriver excited by a single tone f 1 for varied voltage (2 V steps)

Although we are increasing the voltage U 1 from 2 V up to 20 V in2 V steps, we observe a slower increase of the displacement(amplitude compression) at frequencies below and above theresonance f S . This effect is mainly caused by the phase relationship

between current i and displacement x where a current maximumcoincides with a reduced Bl(x) giving less excitation force F=Bl(x)i to the fundamental component. Excitation tones one octave aboveresonance may cause an unstable behavior at high voltages U 1 which is typical for the electrodynamical motor. Even if the rest

position of the coil is well centered in a symmetrical Bl(x) curve,the coil has the tendency to slide down on both slopes of Bl . In thefrequency range where we find significant amplitude compressionsome of the provided energy is transformed into higher-order harmonics. Fig. 7 shows the total harmonic distortion d t in the

A

mm


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radiated sound pressure versus frequency f of the excitation tonefor varied amplitude U 1.

KLIPPEL

0 10 20 30 40 50 60 70 80 90

100

d t

%

Frequency f1 [Hz]

100 1000

Voltage U 1

20

Fig. 7: Total harmonic distortion in the radiated sound pressure of the fictitious driver excited by a single tone f 1 for varied voltage (2V steps)

The harmonic distortion is maximal at excitation frequencies below resonance. This is not only caused by the low excitationforce F=Bl(x)i due to the coincidence of current maximum and

Bl(x) minimum but more by the lowpass characteristic in theradiation of the frequency components below f s.At the resonance frequency there is a pronounced minimum andthe total harmonic distortion measurement has a blind spot for detecting Bl(x)-nonlinearity. However, there is a second maximumapproximately one octave above resonance where we still have

high amplitudes of current and displacement but the phaserelationship between them gives less optimal excitation. At higher frequencies f > 10f s where the amplitude of displacement getssmall the harmonic distortion becomes negligible. This is typicalfor any driver with Bl(x) nonlinearity.

3 A new Performance-based method

Although the current method for assessing X max based on theharmonics distortion measurement fails, the general ideas of thisapproach are still interesting:

Derive X max from drivers performance Dispense with a physical driver model Use standard measurement equipment Keep procedure simple and fast.

3.1 Critical Distortion Measurements

A single tone is a very popular stimulus in distortion measurements because it can be easily generated, and the measured harmonicdistortion can be presented in relation to the excitation frequency.These results represent quite well the total distortion produced bymore complex audio signals as long as the transfer systemcomprises only static non-linearities imbedded in linear systemswith an almost flat amplitude response. For example the limiting of a power amplifier can be modeled by a memory-less system. Herea measurement with a single tone is adequate and the harmonicdistortion has some meaning for a music signal of the sameamplitude.

The dominant non-linearities in electrodynamic transducer are the parameters varying with voice coil displacement. The displacement x is a low-pass filtered signal; also, the other state variables such ascurrent i, velocity v have a different spectral characteristic. In the

nonlinear terms in the differential equation such as the excitationforce F=Bl(x)i , the time signals are multiplied with each other and produce distortion components at all combinations of the inputfrequencies. The instantaneous spectrum of current, displacementand velocity determines the spectral characteristics of the distortionin the output signal. The results of a harmonic distortionmeasurements based on a single tone stimulus are not sufficient to

predict the distortion of the transducer generated by a morecomplex excitation signal.

3.2 Two-tone excitation signalMeasurements of intermodulation components are thereforerequired to get more meaningful results. There are many ways for

performing such measurements [5]. Usually, a multi-tone stimulusis used comprising two or more components. An extensive number of excitation tones might represent an audio signal quite well, butalso produce a lot of data, which have to be interpreted [6]. Thecurrent IEC standard 60268, however, provides a more practicalapproach. A two-tone signal will provide us with the mostimportant information if the frequencies f 1 and f 2 of the first andsecond excitation tones are selected carefully. Since the dominantnonlinearities of most common transducers are related withdisplacement, we have to use the first tone f 1 for generating somevoice coil displacement. Since this tone should be close to theresonance frequency, we may call f 1 the bass tone. The second tone

f 2 may represent any higher frequency component in the pass bandof transducer. Hence we will call it voice tone. Fig. 8 shows thesound pressure spectrum of the fictitious driver excited with a two-tone stimulus represented as bold lines.

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85

90

100

0 250 500 750 1000 1250 1500 1750 2000 2250 2500

Sound pressure level in far field (SPL far)

dB

Frequency [Hz]

Distortion Fundamental

Fig. 8: Spectrum of radiated sound pressure signal of the fictitiousdriver excited by two tones f 1=f s and f 2= 980 Hz at U 1=U 2= 20 V rms .

The thin lines in the SPL spectrum in Fig. 8 are the harmoniccomponents at multiple frequencies of f 1 and the difference andsummed-tone intermodulation at f 2-k*f 1 and f 2+k*f 1, respectively,centered around the voice tone f 2. Usually all higher-order components decrease rapidly with rising order k . Thus, the IECstandard 60268 considers only the low-order componentssummarized as second-order modulation distortion


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%100*)(

)()(

2

12122 f P

f f P f f P d

++=

and third-order intermodulation distortion

%100*)(

)2()2(2

12123 f P

f f P f f P d ++=

referred to the amplitude of the voice tone f 2.Although the amplitudes of both excitation tones are equal, theSPL fundamental f 2 in Fig. 8 is more than 20 dB lower than thefundamental at f 1=f s. The amplitude compression of the voice tone

f 2 is shown more clearly in the Fig. 9 where the SPL of bothfundamentals is displayed versus terminal voltage U 1=U 2.

KLIPPEL

50 55 60 65 70 75 80 85 90 95

10

dB

Voltage U 1 = U 2 [V]

P(f 2)

P(f 1)

1 0.1

Fig. 9: Amplitude of fundamental sound pressure component for atwo-tone excitation signal at f 1=f s and f 2= 780 Hz versus inputvoltage U 1=U 2.

For terminal input voltages below 2 V where the peak displacement is below 1.5 mm we have a linear relationship

between input and output amplitude. At higher voltages the SPL of the voice tone stagnates because the coil will leave the gap for most of the time and the effective excitation of f 2 will not rise.Please note that if we measure the voice tone f 2 without the basstone f 1, we will have almost no amplitude compression.

Fig. 10 shows the third-order modulation distortion according tothe IEC standard versus frequency f 2 while the bass tone is fixed tothe resonance frequency f 1=f s. The voltage U 1=U 2 is increased by 2V steps.

KLIPPEL

25

50

75

100

125

200 40001000 frequency f 2 [Hz]

400

%

Voltage

Fig. 10: Third-order intermodulation distortion in the radiatedsound pressure response for two-tone excitation comprising avariable tone f 2 and a fixed tone f 1=f s with varied voltage ( 2Vsteps)

Neglecting some interferences between harmonic andintermodulation components at multiples of f 1 the intermodulationcomponents d 2 and d 3 are almost constant for f 2 > 3 f s. This istypical for drivers with dominant Bl(x) nonlinearity. Theintermodulation distortion d 2 and d 3 rise monotonically with theterminal voltage. For a terminal voltage of U 1=U 2= 1.3 V rms, thefictitious driver produces already d 3= 10 %. This corresponds to a

peak displacement X max = 1.2 mm.

3.3 Measurement SetupThe loudspeaker modeling and numerical simulation shows thatthe combination of a harmonic and intermodulation distortionmeasurement provides essential information for defining X max moreclearly. A two-tone signal with fixed frequencies is an optimalstimulus that can be produced simply by two sinusoidal generators.Performing a series of measurements with varied frequencies f 1 and

f 2 is not necessary but variations of the terminal voltage arerequired. This is a main difference from measurements of the linear transfer function where we expect the same response from thesystem at low and high amplitudes. For the bass tone f 1 theresonance frequency f s is a distinct frequency giving high voicecoil displacements, low input current and sufficient sound pressurelevel output at the lower end of the transfer band. The frequency of the voice tone f 2 is apparently not critical. The frequency of voicetone f 2 should be much higher than f 1 so it generates not muchdisplacement but significant input current. To avoid interferenceswith harmonics of the fundamental frequency f 1, a fractional ratio

f 2 /f 1 =5.5 may be used between both tones. However, the IECstandard recommends f 2 > 8f 1 making the second order modulation

distortion d 2 more sensitive to Doppler effect. The standard alsosuggest an amplitude ratio of U 1=4*U 2. Using the same amplitudefor both tones U 1=U 2 would give similar values of the modulationdistortion d 2 and d 3 but a much better signal to noise ratio for theintermodulation distortion, and would allow us to compare theharmonics of the bass and voice tone with each other. Althoughthese modifications bring some advantages we will stay withrecommended methods of IEC 60268.To assess the output distortion we have to monitor the sound

pressure signal. We recommend setting the microphone in the near field of the driver, close to the diaphragm, to avoid a free fieldacoustical environment. To find X max we also have to measure thevoice coil displacement precisely. A displacement meter is aindispensable tool for driver design. Laser sensors based on thetriangulation principle are not much more expensive than amicrophone and can also measure the DC displacement


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component. It is also recommended to monitor the input current byusing a shunt or current sensor.A FFT analysis of the measured state variable y(t) provides thefundamentals Y(f 1 ), Y(f 2 ), harmonics of bass and voice tone Y(kf 1 )

and Y(kf 2 ), and the sum- and difference-tone intermodulationY(f 2kf 1 ) of order k . In addition to the distortion measures d t , d 2 andd 3, we recommend also calculating separated second-order harmonic distortion

%100*)(...)3()2()(

)2(22222 kf Y f Y f Y f Y

f Y d h ++++

=

and third-order harmonic distortion

%100*)(...)3()2()(

)3(22223 kf Y f Y f Y f Y

f Y d h ++++

=

These measures show the effect of symmetrical and asymmetrical parameter variation more analytically.

3.4 Dominant Source of DistortionApplying the methods of IEC 60268 to the spectral components of sound pressure, displacement and current, we get a set of distortionmeasures described in Table II.

DISTORTIONMEASURES

INTERPRETATION

X DC The DC part in the displacement is generated dynamically by signal rectification due to parameter asymmetries. The DC part X DC generated by the two-tone signal is mainly caused by suspension asymmetries shifting the coil always towards theminimum of the nonlinear stiffness curve K ms(x).

d h2,f1 The second-order harmonic distortion considering sound pressure component P(2f s ) is a good indicator for asymmetricalstiffness K ms(x). It also reflects some effects of asymmetrical force factor Bl(x). It is insensitive to the nonlinear inductance

Le(x) because the amplitude of the current is low at the resonance.

d h3,f1 The third-order harmonic distortion considering sound pressure component P(3f s ) is a good indicator for symmetricalvariations of the stiffness K ms(x). It partly reflects the symmetrical variations of force factor Bl(x). It is insensitive to thenonlinear inductance Le(x) because the amplitude of the current is low at the resonance.

d 2 The second-order intermodulation distortion considering sound pressure components P(f 2f 1 ) is a good indicator for asymmetrical variations of inductance Le(x), of force factor Bl(x) and Doppler effect. The effect of asymmetries instiffness K ms(x) is negligible.

d 3 The third-order intermodulation distortion in sound pressure considering P(f 22f 1 ) is a good indicator for symmetricalvariations of force factor Bl(x) due to the limited voice coil height. The effects of the other nonlinearities such asinductance L3(x), stiffness K ms(x) and Doppler effect are negligible.

d 2,i The second-order intermodulation distortion considering current components I(f 2f 1 ) is a good indicator for asymmetricalvariations of inductance L3(x). The effect of the other nonlinearities such as force factor Bl(x), stiffness K ms(x) and Doppler effect are negligible.

d 3,i The third-order intermodulation distortion considering current components I(f 2f 1 ) is a good indicator for symmetricalvariations of inductance L3(x). The effects of the other nonlinearities such force factor Bl(x), stiffness K ms(x), Doppler andradiation are negligible.

d h2,f2 The second-order harmonic distortion considering sound pressure component P(2f 2 ) is a good indicator for the reluctanceforce due to asymmetrical inductance Le(x). It also reveals flux modulation due to the asymmetrical variation of the Bl(i) versus voice coil current i and other non-linearities in the driver (partial vibration in the diaphragm, etc.). Thismeasurement is insensitive to variations of force factor Bl(x) and stiffness K

ms(x) versus displacement and Doppler effect.

d h3,f2 The third-order harmonic distortion in sound pressure at P(3f 2 ) is good indicator for flux modulation due to thesymmetrical variation of the Bl(i) versus voice coil current i. It also reflects some other minor nonlinearities in the driver (partial vibration in the diaphragm, etc.). This measurement is insensitive to variations of force factor Bl(x), stiffness

K ms(x) and inductance Le(x) versus displacement and Doppler effect.

Table II: Distortion measures based on two-tone signal


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Physical Cause X DC d h2,f1 d h3,f1 d 2 d 3 d 2,i d 3,i d h2,f2 d h3,f2

Coil offset and asymmetry of Bl(x)

x xCoil height x x

Asymmetry in suspension x x

Symmetrical limiting of suspension x

Asymmetry in L e(x) x x

Symmetrical variation in L e(x) x x

Reluctance Force x

Flux modulation x x

Doppler x

Nonlinear Radiation x x

Partial Cone Vibration x x

Table III: Relationship between nonlinearities and distortion measures (bold symbols represent significant distortion)

The distortion measurements listed in Table II give some cluesabout the physical causes that limit peak displacement X max. Therelationships are represented by crosses in Table III . The dominantnon-linearities caused by the force factor Bl(x), inductance Le(x) and compliance C ms(x) of the mechanical suspension and theDoppler effect may produce substantial values of distortion(greater 5 %). They are the limiting factors of X max in commontransducers and are emphasized by bold crosses. The variation of the radiation conditions cause relatively small distortion for frequencies below 1 kHz. The other nonlinearities such as fluxmodulation and partial cone vibration produce much less distortionin common transducer.The usage of Table III is quite simple. A driver havingsignificant values of d h2,f1 and d 2 suffers from Bl -asymmetry caused

by a coil offset or field geometry. If a high value of d 2 coincideswith significant d 2,i in the input current then the asymmetry of theinductance Le(x) should be reduced by using a short cut ring or copper cap. The Doppler effect can be easily identified by getting

a high value of d 2 coupled with a low value of d 2,i. Anasymmetrical suspension can easily be detected by high values of d h2,f1 and significant X DC while the other second-order distortion aresmall.

3.5 New X max DefinitionSummarizing the considerations we may suggest the wording of the X max definition:

X max is the voice-coil peak displacement at which the maximal value of either the total harmonic distortion d t or the 2 nd order modulation distortion d 2 or the 3 rd -order modulation distortion d 3 in the radiated sound pressure is equal to a defined threshold (d=10 %). The driver is operated in free air and is excited by the

linear superposition of a first tone at the resonance frequency f 1=f s and a second tone f 2=8.5 f s with an amplitude ratio of 4:1. Thetotal harmonic distortion d t assesses the harmonics of f 1 and themodulation distortions are measured according to IEC 60268 inthe near field of the driver. Manufacturer shall state X max , thedominant type of distortion (d t , d 2 or d 3 ) and the value of thethreshold d used.

3.6 Practical Use

1. Measure the resonance frequency f s of the driver.2. Excite the driver under voltage drive with a two-tone

signal at f 1=f s and f 2=8.5 f s with an amplitude ratio of 4:1.

3. Perform a series of measurement while increase theinput amplitude and measure the peak voice coildisplacement and the sound pressure in the near field of

the driver. Perform a spectral analysis of the sound pressure signal and determine the total harmonicdistortion and intermodulation distortion according IEC60268.

4. Search for the minimal value of the peak displacementwhere either d t, d2 or d 3 are equal to the threshold d .

5. State the peak displacement X max, and the type of distortion limiting the excursion.

For example, a statementXmax= 3.8 mm @ d 2=10 % (d t, d3 < 10 %)

means that a driver provides a maximal peak displacement of X max=3.8 mm where the 2 nd-order modulation distortion isdominant and produce the threshold of 10 % distortion. Thisstatement implies that the total harmonic distortion and 3 rd-order distortion are less than 10 % which can be added in parenthesis


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optionally. Thus the suspension and the voice-coil height are mostlikely not the limiting factors for the excursion of this driver.

4 Parameter-based Method

Although the performance-based method gives some indicationabout the dominant source of distortion this approach fails inassessing the limiting factors of each nonlinearity quantitatively.This information is required when the driver designer would like toimprove the maximal output of the driver while keeping the costand other parameters constant. The system designer also needs thisdata to select a driver which produces distortion at X max that areacceptable for his particular application (subwoofer, woofer or full-band system). The parameter-based method provides aseparate value of maximal displacement for each driver nonlinearity which is of practical interest. To avoid any confusionwith the performance-based method these values are called

Displacement limits . The nonlinearity with the smallest value willlimit the peak displacement of the driver finally. The parameter-

based method also uses threshold which should be definedconsistent with the thresholds in the current X max definition to

provide comparable results.

4.1 Displacement Limits due to Driver NonlinearitiesThe maximal voice coil displacement is limited by at least threefactors

1. Excessive decrease of mechanical compliance of themechanical suspension (mainly caused by the naturallimiting of the spider)

2. Voice coil excursion capability (mainly limited byhitting the back plate)

3. Excessive, subjectively unpleasant, signal distortion inthe sound pressure output depending on speaker nonlinearities, intended application, nature of excitationsignal and audible acuity of the listener

These limiting factors may be represented by separate

displacement limits X C represents mechanical loading imposed tosuspension and tolerable distortion due to C ms(x) nonlinearity,

X clip represents free moving range without clipping, X Bl represents tolerable distortion due to Bl(x)-

nonlinearity, X L represents tolerable distortion due to Le(x), L2(x) and

R2(x) nonlinearity, X D represents tolerable distortion due to Doppler

nonlinearity.

4.1.1 Displacement Limit X C The maximal displacement related to the critical mechanical strainof suspension may be obtained from the nonlinear stiffnesscharacteristic K ms(x) or from its counterpart, the compliancecharacteristic C ms(x). Clark [7] suggested to evaluate the variationof this nonlinear parameters. Following his proposal, we introducea minimal compliance ratio

%100*)0()(

min)(min

=


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which is the ratio of the maximal variation of the electricalimpedance at frequency f 2 within the working range -X L < x < X L and the impedance at the rest position x=0.To keep the parameter-based method consistent with the

performance-based method, the frequency f 2= 8.5 f s is coupled tothe resonance frequency f s and the impedance can be approximated by

222

22222 )()(

)()()(),(

s x L x R s x L x R

s x L R f x Z eee +++

where s2=2 f 2 j.

4.1.5 Displacement Limit X D The peak displacement X D considering the audibility of theDoppler effect can be calculated analytically using the simpleequation

2

2770 f

d X peak =

presented by Beers and Belar [9], using the peak displacement X peak in mm, the second-order modulation distortion d 2 in percentaccording to IEC 60268 and the frequency f 2 of the modulatedvoice tone. To keep the definition of X D consistent with the

performance base method we set f 2=8.5 f s and use the distortionthreshold d (d=10%) giving a displacement limit due to Doppler

s D f

d X

5.90=

where X D is in mm and f s is in Hz.

4.2 Practical Use

1. Measure the small signal parameters such as resonancefrequency f s, DC voice coil resistance Re , resistance

R2(0) and inductance L2(0) at x=0 .2. Measure the nonlinear characteristics compliance

C ms(x), force factor Bl(x), and inductance Le(x) versus

displacement x. Listen for excessive distortion andassign X clip=X peak in case of mechanical clipping.3. Determine the peak displacement X c , X Bl , X L and X D by

using the nonlinear characteristics and thresholds for C min , Bmin , Z max and d .

4. State the displacement limits X c , X Bl , X L , X clip and X D together with the thresholds C min , Bmin , Z max and d used.

5 Definition of ThresholdsBoth the peak displacement X max from the performance-basedmethod and the displacement limits X c , X Bl , X L , X clip from the

parameter-based method depend on thresholds. These thresholdsshould consider the audibility of the distortion components, themaximal mechanical load and should lead to comparable results in

both methods.

5.1 AudibilityThe new performance-based method uses the old distortionthreshold d = 10 % for the maximal harmonic distortion. Thisvalue is also applied for the second- and third-order intermodulation distortion. At current time there are no better arguments for using other values. The audibility of the nonlinear distortion generated by loudspeakers depend on the followingfactors:

Linear driver parameters (resonance frequency f s andloss factor Q ts)

Driver nonlinearities ( Bl(x), Le(x), Cms(x) and Doppler) System application (crossover frequency, type of

enclosure) Excitation signal (Nature, bandwidth, spectral and

temporal complexity)

Audible acuity of a listener For example, a suspension nonlinearity produces distortionsconfined to frequencies about the resonance. Contrary, Bl(x) and

Le(x) nonlinearities produce substantial intermodulation throughout

audio band which might be tolerable in subwoofer applications.Thus the evaluation of the nonlinear distortion is a complex issue.Digital transducer modeling gives new possibilities for combiningsubjective and objective investigations by operating theloudspeaker under normal conditions and using ordinary music or any other signal as stimulus. Auralization techniques [10] are the

basis for systematic listening tests providing more reliable data inthe near future.

5.2 Relationship between ThresholdsThe thresholds C min , , Bl min, Z max and d used in the parameterbasedapproach should be consistent with the distortion thresholds of

performance-based approach. Numerical techniques based on theloudspeaker model allows to simulate the sound pressure outputfor some typical shapes of driver nonlinearities and to calculate

parameter ratio Bmin ,C min and Z max that correspond with distortionthreshold d=10% as shown in Table IV. A short and long voicecoil overhang is simulated by a power series expansion of Bl(x)using a quadratic and a fourth-order term, respectively. We assumethat there are no asymmetries in the Bl(x) characteristic. Anonlinear compliance C ms(x) having a quadratic term represents a

progressive spider. The fourth-order term describes thesymmetrical limiting of the surround. A severe asymmetry such ascaused by cup spider can be modeled by a power series of C ms(x)truncated after the linear term. A typical inductance characteristiccan also be approximated by a linear power series expansion.

Example Nonlinear Parameter

Parameter Threshold

DistortionThreshold

Motor withEqual-length

configuration

Bl(x)=b 0+b 2 x2 Bl min 82 % d 3= 10 %

Motor withlarge coiloverhang

Bl(x)=b 0+b 4 x4 Bl min 82 % d 3= 10 %

Progressivespider

C ms(x)=c 0+c 2 x2 C min 74 % d t = 10 %

Linear spider with limitingsurround

C ms(x)=c 0+c 4 x4 C min 77 % d t = 10 %

Asymmetry insuspension

C ms(x)=c 0+c 1 x C min 78 % d t = 10 %

Typicalinductancecharacteristic

Le(x)=l 0+l 1 x Z max 10 % d 2=10%

Table IV: Minimal parameter variation generating 10 % distortionin the radiated sound pressure.

A second-order and fourth-order nonlinearity produces 10 %distortion at similar values of the parameter variation (about 82 %for Bl(x) and about 75 % for C ms(x)). However, a higher order non-linearity will produce much less distortion at lower displacement |X| < X max than a parabola shaped curve. Thus, the nonlinear

parameters themselves are required to predict the distortion versusamplitude, and to explain the benefit of a motor with a larger voicecoil overhang over an equal-length configuration.

5.3 Admissible Mechanical Load The threshold C min=75% producing d t=10% distortion seem to berelatively high compared with the mechanical load admissible tomost drivers. In the Distortion Analyzer system, a minimal


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compliance ratio C min = 50 % is used as a default protection parameter. Most common suspension systems will stand variationdown to a C min= 20 % for some period of time without causing anydamage.

It is also possible that manufacturers define admissible thresholdsthat seem proper for their particular products and specify thesevalues as measurement conditions along with the displacementlimits.

6 Practical Examples

Both the performance-based and parameter-based methods will beapplied to two real drivers to illustrate both techniques. The firstdriver A has an extremely long voice coil coupled with a limitedsuspension. By contrast, the second driver B uses a short coil witha very linear suspension. The nonlinear parameters are measureddynamically by using the Distortion Analyser.

KLIPPEL

0,0

0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5

-4 -3 -2 -1 0 1 2 3 4

[N/A]

>

Bl(X) Bl (-X)

Fig. 12: Force factor Bl(x) versus voice coil displacement x of driver A

KLIPPEL

0,00

0,25

0,50

0,75

1,00

-4 -3 -2 -1 0 1 2 3 4

[mm/N]

>

Cms(X) Cms (-X)

Fig. 13: Compliance C ms(x) of the mechanical suspension versusvoice coil displacement of driver A

KLIPPEL

0,00

0,25

0,50

0,75

1,00

-4 -3 -2 -1 0 1 2 3 4

[mH]

>

Le(x)

Fig. 14: Inductance Le(x) versus voice coil displacement x of driver A

6.1 Driver AThe force factor Bl(x) in Fig. 12 remains almost constant over themeasured range producing low modulation distortion. Consideringa limit of Bmin = 82 %, the admissible peak displacement X B is

beyond 4 mm. Apparently, the magnet field geometry issymmetrical and the coil is at the optimal rest position.However, the compliance C ms(x) in Fig. 13 has a asymmetricalcharacteristic becoming obvious by comparing the regular curveC ms(x) with the mirror curve C ms(-x) presented as dotted line.Considering a limit value of C min=75 %, the admissible peak displacement X C is 2 mm. Due to the asymmetry, the suspensionlimits the excursion only at negative displacement.Fig. 14 shows the asymmetric characteristic of the inductance

Le(x), which is typical for a motor without shortcut ring or copper cap. Considering the resonance frequency f s= 49 Hz, a DCresistance Re = 6.8 and the limit value Z max = 10 %, theadmissible peak displacement X L exceeds the measured range of 4mm.The admissible peak displacement X D producing 10 % modulationdistortion is about 18 mm.Searching for the minimum between the separate peak displacements X B , X C , X L , X D, clearly the suspension limits themaximal displacement to 2 mm approximately.Using the new performance-based method the total harmonicdistortion d t , the second- and third-order modulation are measuredversus peak displacement and presented in Fig. 15.


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0 1 2 3 4 5 60 5

10 15 20 25 30 35 40

harmonic

3 rd modulation

2nd modulation

Peak Displacement [mm]

%

Fig. 15: Total harmonic distortion d t ,(solid line), second-order modulation d 2 (dashed line) and third-order modulation distortiond 3 (dotted line) versus peak displacement x of driver A

Since the suspension is the limiting factor, the total harmonicdistortion dominates and exceeds the 10 % limit at X max= 2.4 mmfirst. The second-order distortion caused by asymmetricalinductance Le(x), Doppler effect and Bl asymmetry cut the 10 %distortion level at 4 mm. The third-order distortion, which isdirectly related to the voice coil height and the symmetrical Blvariation, is far below 10 % up to 6 mm displacement.Table V summarizes the other distortion measures determined at aU 1=U 2=U 10% = 3. 4 V rms giving a peak displacement of X max= 2.4mm.

Speaker A Speaker B

f 1= f s 50 Hz 44 Hz

f 2 =8f 1 400 Hz 360 Hz

U @ d=10 % 3.4 V rms 1.87 V rms

X DC 0.31 mm 0.03 mm

d h2,f1 8.7 % 4.3 %

d h3,f1 5.4 % 3.6 %

d t 10 % 5.7 %

d 2 7.9 % 8.4 %

d 3 1.5 % 10 %

d 2,i 7.7 % 3.3 %

d 3,i 0.5 % 1.2 %

d h2,f2 0.58 % 0.4 %

d h3,f2 0.34 % 0.4 %

X max @ d= 10% 2.4 mm 2.1 mm

Table V: Results of the performance-based method

The second-order harmonic distortion d h2,f1 = 8.7 % of the bass tone f 1 dominates the third-order harmonic d h3,f1 = 5.4 % due to thesubstantial asymmetry of the suspension. The positive DC-displacement generated by rectification of the bass tone shifts thecoil in positive direction where the compliance is maximal. Thus,improving the symmetry of the curve will give more X max.The second-order distortion d 2 in sound pressure and d 2(I) in currentare in the same order of magnitude, indicating that the inductanceasymmetry is the physical source while the contribution of Bl-asymmetry and Doppler is much smaller. The harmonic distortionsof the voice tone reveal the effect of nonlinearities that are relatedto voice coil current or mechanical stress in the diaphragm.However, the distortion measures d h2,f2 , d h3,f2 , d 3,i are as usual below1 %, which can be neglected in comparison to the dominantnonlinearities.The most important information may be stated by themanufacturer:

X max = 2.4 mm @ d t =10 % (d 2 , d 3 < 10 %) X C = 2 mm @ C min=75 % X B > 4 mm @ Bl min=82 % X L > 4 mm @ Z max = 10% X D = 18 mm @ d 2 = 10 %

6.2 Driver B

For a second speaker B we measured the following parameters


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X max = 2.1 mm @ d 3 =10 % (d 2 , d t < 10 %) X C > 4 mm @ C min=75 % X B = 1.8 mm @ Bl min=82 % X L > 4 mm @ Z max = 10 % X D = 20.5 mm @ d 2 = 10 %.

In contrast to speaker A we find dominant third-order intermodulation limiting the peak displacement Xmax. Itcorresponds with dominant force factor nonlinearity causing thelowest displacement limit X B= 1.8 mm. The suspension, theinductance and the Doppler effect give much more excursioncapabilities that cannot be used.

KLIPPEL

0,0

0,5

1,0

1,5

2,0 2,5

3,0

3,5

-4 -3 -2 -1 -0 1 2 3 4

N/A

>

Bl(X) Bl (-X)

Fig. 16: Force factor Bl(x) versus voice coil displacement of driver B

The force factor Bl(x) as displayed in Fig. 16 reveals a short voice

coil with low overhang. Such types of speakers are sensitive to anoffset of the coil. Since the optimal rest position is 0.8 mm insidewe have an asymmetrical characteristic. Considering the limit of

Bmin= 82 % for the decay of Bl(x), we get a peak displacement of X B = 1.8 mm related to the motor capabilities.The compliance C ms(x) in Fig. 17 reveals a very linear suspension.The C ms(x) stays in the measured range above C min= 75 % ( X C > 4mm).Similar to driver A, the inductance Le(x) in Fig. 18 has the typicalshape with a maximum at negative displacement. However, theabsolute value of inductance is less than a third of driver A.Considering the resonance frequency f s= 44 Hz and the DCresistance Re = 3.8 and the limit value of Z max = 10 % the peak displacement X L exceeds 4 mm.Considering the Doppler effect, we get a peak displacement X D =20.5 mm.

KLIPPEL

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,80,9

-4 -3 -2 -1 -0 1 2 3 4

mm/N

>

Cms(X) Cms (-X)

Fig. 17: Compliance C ms(x) of the mechanical suspension versusvoice coil displacement x of driver B

KLIPPEL

0,00

0,05

0,10

0,15

0,20

0,25

0,30

-4 -3 -2 -1 -0 1 2 3 4

[mH

>

Le(X)

Fig. 18: Inductance Le(x) versus voice coil displacement x of driver B

Despite the asymmetry of the Bl(x)-curve which is not consideredin the definition of the threshold Bl min , the results of the parameter-

based approach agree well with the results of the performance- based method. Fig. 19 shows the distortion d t , d 2 and d 3 versus peak displacement X peak measured with the two-tone signal.


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0 1 2 3 4 5 6 70 10 20 30 40 50 60

70 80

%

Peak Displacement mm

2nd modulation

3 rd modulation

Harmonic

Fig. 19: Total harmonic distortion d t (solid line), second-order modulation d 2 (dashed line) and third-order modulation distortiond 3 (dotted line) versus peak displacement x of driver B

Above 2 mm displacement the third-order modulation distortion d 3 becomes dominant and reaches the threshold of 10 % at X max = 2.1mm. Although the parameter-based method does not consider theshape of the nonlinear characteristic but only the crossing point at82 % variation, we get similar results for X Bl .According to Table III significant values of d 3 show that thesymmetrical Bl-variation due to a short coil limits the X max.Considering the total harmonic distortion d t only we would getmuch higher peak displacement X max= 7 mm but at that X max anyvoice tone will produce about d 3= 70 % modulation distortion.The second-order modulation distortion d 2 are twice of theharmonics. Additional measurement of the second-order modulation d 2(I) in input current gives us more information aboutthe source. Since d 2= 6.8 % in sound pressure is significant higher than d 2(I) in current the force factor asymmetry gives a significantcontribution.

7 Summary

The critical review of the numerical simulation on a fictitiousloudspeaker and practical measurements on real loudspeakersshow that the method in AES2-1984 does not provide a clear anduseful definition of X max. This is mainly caused by someambiguities in the wording and more importantly by usingassumptions, which are not valid in theory and practice. Clearly,the measurement of harmonic distortion is not sufficient for assessing all important aspects of the large signal performance asemphasized by Voishvillo [5]. Nonlinearities inherent intransducers such as force factor Bl(x), inductance Le(x) andDoppler produce significant modulation distortion. The currentIEC standard 60268 provides all of the methods required for assessing these kinds of distortion and for defining X max moreclearly and reliably. The new definition is based on a two-tonemeasurement that can be accomplished with straightforwardequipment. The resulting distortion measures are also valuable for transducer diagnostics to improve the driver design or select theoptimal driver for the particular application.The second part of the paper addressed an alternative method for assessing separate displacement limits closely related with thenonlinear driver parameters. In contrast to the performance-basedapproach, which measures some effects such as distortion of thenonlinear systems, the parameter-based method refers to the

physical causes. The nonlinear curves and other linear parametersare summarized to a few numbers describing the limiting effect of each driver nonlinearity (Bl(x), Cms(x), L e(x) and Doppler). This

is a substantial data reduction where some particularities of thenonlinear curves are neglected. Despite the simplifications made in

both methods the minimal value of the displacement limits X Bl XL,XC, X D is comparable with the peak displacement X max derived

from distortion measurement. Numerical tools are available for transforming and comparing the results of both methods.Do both methods compete with each other and will the new

parameter-based method eventually replace the performance-basedmethod? I don't think so. It is a good idea to state X max based ondistortion measurements because it can easily be verified by simpleequipment. The displacement limits X B , X C , X L , X D give additionalinformation about the driver which are important for the systemdesign.The main target of this paper was the development of a framework for assessing the peak displacement more reliably. Both methods

presented are still flexible by changing the thresholds used. Thiscan be easily accomplished by an agreement between driver andsystem manufacturer considering special requirements. More longterm testing of loudspeakers with instantaneous parameter monitoring and systematic listening tests using a digital transducer model will provide more information about assessable load andimpact on sound quality. In the meantime the "traditional"threshold of 10 % distortion is a good starting point.

8 Acknowledgment

This paper is the result of many fertile discussions that I had withAlexander Voishvillo, David Clark and others.

9 References

[1] M. Gander, "Dynamic Linearity and Power Compression inMoving-Coil Loudspeakers," J. Audio Eng. Soc. vol. 34, pp. 895 -904 (1986 November).

[2] R. Small, "Direct-Radiator Loudspeaker System Analysis, J. Audio Eng. Soc . (June 1972).

[3] D.B. Keele, "Direct Low-Frequency Driver Synthesis fromSystem Specifications J. Audio Eng. Soc. vol. 30, pp. 800 - 814(1982 November).

[4] W. Klippel, "Prediction of Speaker Performance at HighAmplitudes," presented at 111th Convention of the AudioEngineering Society, 2001 September 2124, New York, NY,USA

[5] A. Voishvillo, "Assessment of Loudspeaker Large SignalPerformance Comparison of Different Testing Methods and

Signals," Contribution to the Task Group SC-04-03-C "LargeSignal Parameters of Low-Frequency Drivers " presented at 111thAES Convention, New York, Nov. /Dec. 2001

[6] E. Czerwinski, et. al., "Multitone Testing of Sound SystemComponents Some Results and Conclusions, J. Audio Eng. Soc. vol. 49, pp. 1011 - 1048 (2001 Nov.).

[7] D. Clark, Precision Measurement of Loudspeaker Parameters, J. Audio Eng. Soc. vol. 45, pp. 129 - 140 (1997 March).

[8] W. Klippel, Distortion Analyzer a New Tool for Assessingand Improving Electrodynamic Transducers, presented at the108th Convention of the Audio Engineering Society, Paris,February 19-22, 2000, preprint 5109


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[9] G.L. Beers and H. Belar, "Frequency-Modulation Distortion inLoudspeakers," J. Audio Eng. Soc . vol. 29, pp. 320 326, (1981May).

[10] W. Klippel, "Speaker Auralization Subjective Evaluation of Nonlinear Distortion," presented at the 110th Convention of theAudio Engineering Society, Amsterdam, 2001 May 12-15, preprint5310.

Assessment of Voice Coil Peak Displacement XMAX 02

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